Analysis 1A exercise sheet 4: Sequences and limits
1. Determine whether or not the following sequences are bounded (justify
your answers):
(a) (an )n∈N where an =
(b) (an )n∈N where an =
(c) (an )n∈N where an =
n+3
n for all n ∈ N,
n
n+2 for all n ∈ N,
n2 +1
n+3 for all n ∈ N,
(d) (an )n∈N where an = −n for all n ∈ N.
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2. Let (xn )(n∈N) be the sequence defined by xn = n+4
for all n ∈ N. Show
that lim xn = 0. This should be answered working directly from the
n→∞
definition (that means not using the sandwich rule or Theorem 4.9
from the lecture notes).
n
3. Let (xn )n∈N be the sequence defined by xn = n+2
for all n ∈ N. Show
that lim xn = 1. This should be answered working directly from the
n→∞
definition (that means not using the sandwich rule or Theorem 4.9
from the lecture notes).
4. Show that the following sequences are divergent:
(a) (an )n∈N where an = 2(−1)n for all n ∈ N.
(b) (an )n∈N where an = 6n for all n ∈ N.
(c) (an )n∈N where an =
n(−1)n +8
n
for all n ∈ N.
5. Let (xn )n∈N be a real valued sequence which is convergent with lim xn =
n→∞
x for some x ∈ R. Let k ∈ N and (yn ) by the real valued sequence
where yn = xn+k for all n ∈ N. Prove that lim yn = x.
n→∞
6. Let (an )n∈N and (bn )n∈N be real valued sequences and a, b ∈ R. Suppose that an ≤ bn for all n ∈ N, lim an = a and lim bn = b. Show
n→∞
n→∞
that a ≤ b. (Remember to show that a ≤ b it suffices to show that for
all > 0, a − b ≤ .)
7. Let (xn )n∈N be a real valued sequence and a ∈ (0, ∞). Show that if
lim |xk+1 − xk | = a then (xn ) is divergent.
k→∞
8. Let (xn )(n∈N) be a sequence of non-negative real numbers and
√suppose
√
that lim xn = x. Show that x ≥ 0 and that lim xn = x. To
n→∞
n→∞
√
√
show that lim xn = x consider the cases where x = 0 and
n→∞
x > 0 separately, for the second case it may help to use that
√ √
√
√
√
( xn − x)( xn + x)
√
√
( xn − x) =
.
√
xn + x
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9. For each of the following real valued sequences (an )n∈N find lim an .
n→∞
Justify your answers, quoting which results from the lectures or earlier
questions you are using.
(a) an =
1
n3 +5
(b) an =
n2 +3
4n2 +7n
(c) an =
√
for all n ∈ N.
for all n ∈ N.
√
n + 1 − n for all n ∈ N.
(d) an = sin n+5n
for all n ∈ N. (NB: You may use that | sin n| ≤ 1
n2
for all n ∈ N without justification.)
10. Let (xn )n∈N be a sequence of real numbers, let x ∈ (−∞, 0) and suppose that lim xn = x. Show that there are only finitely many k ∈ N
n→∞
where xk ≥ 0.
11. Let (an )n∈N and (bn )n∈N be sequences of real numbers. Suppose that
(bn ) is bounded and lim an = 0. Show that lim an bn = 0.
n→∞
n→∞
12. Let (an )n∈N and (bn )n∈N be sequences of real numbers and a ∈ R\{0}.
Suppose that (bn )n∈N is divergent and lim an = a. Show that the
n→∞
sequence (an bn ) is divergent.
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